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Standing waves and global existence for nonlinear wave equations with potential, strong, and nonlinear damping terms
Boundary Value Problems volume 2014, Article number: 144 (2014)
This paper is concerned with the Cauchy problem of nonlinear wave equations with potential, strong, and nonlinear damping terms. Firstly, by using variational calculus and compactness lemma, the existence of standing waves of the ground states is obtained. Then the instability of the standing wave is shown by applying potential-well arguments and concavity methods. Finally, we show how small the initial data are for the global solutions to exist.
Consider the Cauchy problem for nonlinear wave equations with potential, strong, and nonlinear damping terms,
where , , , ,
For the case of linear damping (, ) and nonlinear sources, Levine  showed that the solutions to (1.1) with negative initial energy blow-up for the abstract version. For the nonlinear damping and source terms (, , , ), the abstract version has been considered by many researchers –. For instance, Georgiev and Todorova  prove that if , a global weak solution exists for any initial data, while if the solution blows up in finite time when the initial energy is sufficiently negative. In , Todorova considers the additional restriction on . Ikehata  considers the solutions of (1.1) with small positive initial energy, using the so-called ‘potential-well’ theory. The case of strong damping (, , ) and nonlinear source terms () has been studied by Gazzola and Squassina in . They prove the global existence of solutions with initial data in the potential well and show that every global solution is uniformly bounded in the natural phase space. Moreover, they prove finite time blow-up for solutions with high energy initial data. However, they do not consider the case of a nonlinear damping term (, , ).
To the best of our knowledge, little work has been carried out on the existence and instability of the standing wave for (1.1). In this paper, we study the existence of a standing wave with ground state (), which is the minimal action solution of the following elliptic equation:
Based on the characterization of the ground state and the local well-posedness theory , we investigate the instability of the standing wave for the Cauchy problem (1.1). Finally, we derive a sufficient condition of global existence of solutions to the Cauchy problem (1.1) by using the relation between initial data and the ground state solution of (1.4). It should be pointed out that these results in the present paper are unknown to (1.1) before.
For simplicity, throughout this paper we denote by and arbitrary positive constants by .
Preliminaries and statement of main results
We define the energy space in the course of nature as
By its definition, is a Hilbert space, continuously embedded in , when endowed with the inner product as follows:
whose associated norm is denoted by . If , then
Throughout this paper, we make the following assumptions on :
From Proposition 2.1, it follows that is the critical case, namely for , a weak solution exists globally in time for any compactly supported initial data; while for , blow-up of the solution to the Cauchy problem (1.1) occurs.
We define the functionals
for , and we define the set
We consider the constrained variational problem
For the Cauchy problem (1.1), we define unstable and stable sets, and , as follows:
The main results of this paper are the following.
There exists such that
(a2) is a ground state solution of (1.4).
From Theorem 2.3, we have the following.
(b1) If, and there existssuch that, then the solutionof the Cauchy problem (1.1) blows up in a finite time.
(b2) If, there existssuch that, then the solutionof the Cauchy problem (1.1) globally exists on. Moreover, for, satisfies
Variational characterization of the ground state
In this section, we prove Theorem 2.3.
The constrained variational problem
is equivalent to
andprovidedas well as.
Let . Since
it follows that
Thus by , one sees that there exists some such that
where uniquely depends on and satisfies
which together with and (3.6) implies that , we have
Therefore, the above estimates lead to
It is easy to see that
It follows that and on .
For any and , let . There exists a such that , and from (3.8) we get
Finally, we prove by showing in terms of the above equivalence. Since , the Sobolev embedding inequality yields
From , it follows that
which together with implies
Therefore, from (3.10), we get
This completes the proof of Lemma 3.1. □
is bounded below on M and.
It follows that on . So is bounded below on . From (2.6) we have . □
Let, forand. Then there exists a unique (depending on) such that. Moreover,
From the definition of , there exists a unique such that . Moreover,
and , we have , . □
Next, we solve the variational problem (2.6).
We first give a compactness lemma in .
Letwhenandwhen. Then the embeddingis compact.
In the following, we prove Theorem 2.3.
Proof of Theorem 2.3
According to Proposition 3.2, we let be a minimizing sequence for (2.6), that is,
For simplicity, we still denote by . So we have
By Lemma 3.4, we have
Since , , we obtain
On the other hand, from (2.3) we have
Since and by Proposition 3.3, we get
Since and , then . Therefore from (3.27), solves the minimization problem
Thus we complete the proof of (a1) of Theorem 2.3.
Next we prove (a2) of Theorem 2.3.
Since is a solution of the problem (3.28), there exists a Lagrange multiplier such that
where denotes the variation of about . By the formula
where denotes the variation of . From (3.29), we get
From , we have
So far, we have completed the proof of Theorem 2.3. □
Blow-up and global existence
In this section, we prove Theorem 2.5.
Let, thenandare invariant under the flow generated by (1.1).
To prove , we need to prove
Similarly, we can show that is also invariant under the flow generated by (1.1). This completes the proof of Lemma 4.1. □
is fulfilled for.
According to Lemma 4.1, we have for . Using the above inequality and the identity (2.3), we get
which completes the proof of Lemma 4.2. □
Next, we prove Theorem 2.5.
Proof of Theorem 2.5
By (2.7), we have .
Firstly, we prove (b1) of Theorem 2.5. From the energy identity we have
for all .
Denoting , we have
Using the Hölder inequality and the interpolation inequality, we obtain
with . From , we have
which together with Lemma 4.1 give
Using the Young inequality and , we have
where the constant is chosen as follows.
Since , we choose the constant so that
This guarantees that . Then, using this choice and Lemma 4.2, we get
If the constant is fixed, we choose the constant such that
Finally, using the inequality (4.6) and Lemmas 4.1 and 4.2 we have
from which one concludes that there exists a such that . Hence, is increasing for (which is the interval of existence). Since , there exists a such that is increasing for . When is large enough, the quantities and are small enough. Otherwise, assume that there is such that for all . By integrating the inequality, we obtain a contradiction with (4.3) and .
Thus in these cases, the quantity
Using the Hölder inequality, we get
from (4.10) we have . Therefore is concave for sufficiently large , and there exists a finite time such that
From assumption on , we obtain
Thus one get and
Now we complete the proof of (b1) of Theorem 2.5.
Next, we prove (b2) of Theorem 2.5.
From (2.7) and Lemma 4.1, we obtain . It follows that satisfies
which will be proved by contradiction. If (4.11) is not true, then we have . Thus there exists such that and
which implies .
On the other hand, for , and (2.3) yield
which is contradictory to Lemma 3.1.
Therefore, by and Lemma 4.1, we have and . Thus
Therefore we have established the bound of in for and thus the solution of (1.1) exists globally on .
Thus, we complete the proof of Theorem 2.5. □
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The author declares that they have no competing interests.
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Cite this article
Huang, W. Standing waves and global existence for nonlinear wave equations with potential, strong, and nonlinear damping terms. Bound Value Probl 2014, 144 (2014). https://doi.org/10.1186/s13661-014-0144-0
- wave equations
- nonlinear damping terms
- strong damping terms
- global existence